Problem 33 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt[3]{2}(4 \sqrt[3]{4}+\sqrt[3]{12}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 8 + 2 \sqrt[3]{3} \).

Step 1: Distribution

To begin solving the expression \( \sqrt[3]{2}(4 \sqrt[3]{4} + \sqrt[3]{12}) \), first distribute \( \sqrt[3]{2} \) to each term within the parentheses. This gives us:\[ \sqrt[3]{2} \times 4 \sqrt[3]{4} + \sqrt[3]{2} \times \sqrt[3]{12} \]

Step 2: Simplify Each Cube Root Term

Next, simplify each resulting term by multiplying the cube roots. For the first term, multiply \( \sqrt[3]{2} \) and \( \sqrt[3]{4} \):\[ 4 \times \sqrt[3]{2 imes 4} = 4 \times \sqrt[3]{8} \]Recall that \( \sqrt[3]{8} = 2 \).For the second term, multiply \( \sqrt[3]{2} \) and \( \sqrt[3]{12} \):\[ \sqrt[3]{2 imes 12} = \sqrt[3]{24} \]

Step 3: Simplify Further

Now simplify the terms:- For \( 4 \times \sqrt[3]{8} \), since \( \sqrt[3]{8} = 2 \), this term simplifies to \( 4 \times 2 = 8 \).- The second term \( \sqrt[3]{24} \) can be simplified because \( 24 = 2^3 \times 3 \), thus \( \sqrt[3]{24} = \sqrt[3]{8 \times 3} = 2 \times \sqrt[3]{3} \).

Step 4: Combine the Terms

Combine the simplified terms from the previous steps:\[ 8 + 2 \sqrt[3]{3} \]Thus, the product and simplification of the expression is complete.

Key Concepts

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Unlike square roots, which deal with power of two, cube roots involve power of three. For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\).
Cube roots are represented by the symbol \(\sqrt[3]{x}\), where \(x\) is the number you're finding the cube root of. In the given exercise, we have expressions like \(\sqrt[3]{2}\), \(\sqrt[3]{4}\), and \(\sqrt[3]{12}\).
When simplifying expressions with cube roots, always check if the number inside the root is a perfect cube, making it easier to simplify. Identifying these numbers will save you time and make your calculations more straightforward.
For not perfect cube numbers, like 12, it can be useful to break them into factors to find smaller cubes inside. For \(\sqrt[3]{12}\), you can see it's made from \(2^2 \times 3\). In general, breaking numbers into their prime factors helps simplify cube roots.

Simplification

Simplification in mathematics means reducing an expression to its simplest form. This can involve combining like terms, reducing fractions, or, as in this exercise, simplifying roots.
Our goal when simplifying expressions with cube roots is to make them as straightforward as possible. This means finding any perfect cubes within the numbers and converting them into whole numbers. For instance, in the step where we have \(4 \times \sqrt[3]{8}\), knowing \(\sqrt[3]{8} = 2\) helps simplify this to \(4 \times 2 = 8\).
Simplification also involves rationalizing the expression. For example, in \(\sqrt[3]{24}\), we identified that \(24 = 2^3 \times 3\), allowing us to take out \(\sqrt[3]{8}\) or \(2\) and leaving us with \(2 \times \sqrt[3]{3}\). This keeps the expression clean and manageable.

Distribution in Algebraic Expressions

Distribution is a key property in algebra that helps in expanding and simplifying expressions. It involves multiplying a single term by each term within a parenthesis. In arithmetic, this property is known as the distributive property and is expressed as \(a(b + c) = ab + ac\).
In this exercise, we use distribution to handle the expression \(\sqrt[3]{2}(4 \sqrt[3]{4} + \sqrt[3]{12})\). By multiplying \(\sqrt[3]{2}\) with each term inside the parentheses individually, we expand the expression:

• Multiplying the first term gives \(\sqrt[3]{2} \times 4 \sqrt[3]{4}\).
• Multiplying the second term yields \(\sqrt[3]{2} \times \sqrt[3]{12}\).

This application of distribution is crucial for simplifying since it breaks down complex expressions into more manageable parts that can be simplified individually. Ensuring you fully distribute each term helps avoid common mistakes and keeps your work organized.